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Does the "field" in physics same as the "field" in mathematics? Which one those is more abstract in Nature?

Abu sayed
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  • "Field" has more than one meaning in mathematics. Could you link to a Wikipedia article or something to clarify which uses of the word you'd like to compare? – DanielSank Aug 23 '16 at 06:24
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    The word field in mathematics is commonly used to denote a ring whose (nonzero) elements form an abelian group under multiplication. A vector field is another type of mathematical object, that shares features with some of the things denoted in physics by the word field (but not all of them). – yuggib Aug 23 '16 at 06:41
  • In mathematics there are hundreds of field examples. Physically its trickier especially when it comes to electromagnetic phenomena. For instance we are told that light is an electromagnetic wave and not made of photons. If you ask what the electromagnetic wave is the term field comes up quite a bit. If you ask what an electric or magnetic field is made of it says photons for either one?? What gives? – Bill Alsept Aug 23 '16 at 07:46
  • @ACuriousMind A real field really gives a physical effect but not one of those answers on the other question could give a physical description of a field. If an electromagnetic field is not made of photons what is it??? – Bill Alsept Aug 23 '16 at 21:28

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The terminology addressing the meaning of field, in physics, tends to change according to the area and the context one is dealing with.

In general a field is a map $p\mapsto T(p)$ that assigns a quantity to each point in the domain of definition of the theory, in order to define the states of the theory itself and their evolution.

  1. In point particle classical mechanics a field is a map $$ t\in \mathbb{R}\to (\mathbf{r}(t),\mathbf{v}(t))\in\mathbb{R^6} $$ giving the position and the velocity of the particle at any time $t$. More generally, for a system of classical particles a field is a map $$ s\in \mathbb{R}\to \gamma(s)\in T^*Q $$ with $T^*Q$ being the phase space as cotangent bundle of the configuration space.

  2. In quantum mechanics a field is a map $$ t\in \mathbb{R}\to |\psi(t)\rangle\in\mathcal{H} $$ that assigns an element of a Hilbert space to any point $t$ in time.

  3. In classical electromagnetism a field is a map $$ (x,t)\in \mathbb{R^4}\to A^{\mu}(x,t)\in\mathbb{R^4} $$ that assigns a vector potential to every point in space and time $(x,t)$. More generally, for gauge theories, a field is a map $$ m\in \mathcal{M}\to T(m)\in\chi(\mathcal{M}) $$ that assigns a tensor to any point $m$ of a smooth manifold $\mathcal{M}$. Likewise for quantum gauge theories, where the above classical gauge fields are integrated in the path integral to give rise to correlation functions.

  4. In thermodynamics a field is a map $$ (X_1,\ldots,X_N)\in\mathbb{R^N}\to S(X_1,\ldots,X_N)\in\mathbb{R} $$ that assigns the entropy given a set of extensive variables $(X_1,\ldots,X_N)$

and many more examples can be provided.

gented
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    Don't get it bad, but it seems that you're just listing functions that are used in physics, calling them "fields" ;-P For example, I would call 1 above the Hamiltonian flow, 2 the quantum evolution map (in Schrödinger picture), 3 the electromagnetic potential, 4 the entropy function. As far as I know, in physics the word "field" is used either to denote a (classical) force field, or the self-adjoint-operator-valued distributions of quantum field theory. – yuggib Aug 23 '16 at 13:48
  • @yuggib Fields are in fact functions, by definition, and in physics they do in fact correspond to the system evolution (that's why physics theories are field theories): why does that surprise you? – gented Aug 23 '16 at 14:06
  • Mostly, I don't understand (or agree with) your use of terminology. You are writing down functions used in physics that are not "physical fields" (and mostly neither "mathematical vector fields") and calling them fields. I would never use the word "field" for the Hamiltonian flow, or the quantum evolution map, or the entropy; and I would also say that it is not just me: your terminology seems quite non-standard. – yuggib Aug 23 '16 at 15:00
  • The Hamiltonian flow is a field with values in $T^Q$ (that in fact contains vectors and forms, if the vectors are what you're looking for, see any textbook in differential geometry), the quantum wave function is* a (complex valued) field (see any textbook in quantum mechanics) and any smooth map from a manifold into its tensor bundle is a field (seen as section of the bundle itself). Physical fields aren’t the ones with the arrows on top, if this is what you have in mind. We can agree that some of them aren't "observables" but surely they are mathematical fields. – gented Aug 23 '16 at 15:24
  • The Hamiltonian flow at most is an integral curve for the Hamiltonian vector field (and not a vector field itself), and the quantum wave fucntion is a vector of the Hilbert space (and not a vector field). In physics, the word field is mostly used for force fields (e.g. gravitational, electromagnetic) and for quantum fields (that are mathematically operator valued distributions). – yuggib Aug 23 '16 at 15:47
  • "the quantum wave fucntion is a vector of the Hilbert space" that's first grade mistake between a function and the value of the function. I don't know what you call the Hamiltonian flow, but in my terminology it's the map that assigns to each time $t$ an element on $T^*Q$. – gented Aug 23 '16 at 15:57
  • The wave function is a square integrable function that represents the "probability amplitude" of the quantum state, or more in general it is a vector of the Hilbert space used to represent the canonical commutation relations in the system, check any standard textbook. And the Hamiltonian flow is not a vector field in geometric terms (the latter is a section of the tangent bundle while the former is a map from the reals to the cotangent bundle). I am not the one making first grade mistakes... – yuggib Aug 23 '16 at 16:39
  • I guess you have an odd idea of what a function is: a function is a map $x\to f(x)$, namely a subset of the cartesian product containing all the pairs $(x, f(x))$. The values of a function are the subset of points $f(x)$. In the example at hand the values of the field are the states $|\psi(t)\rangle \in \mathcal{H}$, but the function is the entire assignment. That then it may be interpreted as probability density I agree, but that's not the point. – gented Aug 23 '16 at 17:36